Existence and non-existence of normalized solutions for a nonlinear fractional Schr\"odinger system

This paper is concerned with a nonlinear fractional Sch\"ordinger system in $\mathbb{{R}}$ with intraspecies interactions $a_{i}>0 \ (i=1,2)$ and interspecies interactions $\beta \in\mathbb{{R}}$. We study this system by solving an associated constrained minimization problem (i.e., $L^2-$norm constrains). Under certain assumptions on the trapping potentials $V_i(x) \ (i=1,2),$ we derive some delicate estimates for the related energy functional and establish a criterion for the existence and non-existence of solutions, in which way several existence results are obtained.